Not your typical car problem

Calculus Level 5

A certain road follows the path of a sine wave, with the equation \(y = \sin \left(\frac {\pi}{6} x\right) \), whose origin represents the starting point of the road. If a car is travelling at this road starting at its origin, at an instantaneous speed of \(\frac { 9x^2}{3x^2 - 1} \sqrt{1+ \frac {\pi^2}{36}\cos^{2}\left( \frac{\pi}{6}x\right)} \) m/s, as the car moves farther, determine the limiting rate (in m/s) as to how the car separates from the origin.


The values of \(x\) and \(y\) are in meters.

Clarification: In the graph above, the car follows the red path.


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